Conversions Related to Decimal System

In this articles we will discuss the following conversions related to the decimal system.

1. Decimal to other systems.
2. Other systems to decimal.

Conversion from Other Systems to Decimal:

The general procedure for conversion from any other systems to decimal is as given below:

• Step 1: Note down the given number.
• Step 2: Write down the weights for different positions.
• Step 3: Multiply each digit in the given number with the corresponding weight to obtain product numbers.
• Step 4: Add all the product numbers to get the decimal equivalent.

Let us follow the procedure to obtain the conversion from binary, octal and hex numbers to decimal.

Convert the binary number 1011.01 into its decimal equivalent.

Step 1: Note down the given number.

            1011.01

Step 2: Write down the weights for different positions.

            1 = 2³  ;  0 = 2²  ;  1 = 2¹  ;  1 = 2⁰  ;  0 = 2^-1  ;  1 = 2^-2

Step 3: Multiply each digit in the given number with the corresponding weight to obtain product numbers.
            1*2³ = 8  ;  0*2² = 0  ;  1*2¹ = 2  ;  1*2⁰ = 1  ;  0*2^-1 = 0  ;  1*2^-2 = 0.25

Step 4: Add all the product numbers to get the decimal equivalent.

            1*2³ + 0*2² + 1*2¹ + 1*2⁰ + 0*2^-1 + 1*2^-2

        =  8 + 0 + 2 + 1 + 0 + 0.25

        =  11.25

Therefore; (1011.01)₂ = (11.25)10

Convert the Octal number (314)₈ into its decimal equivalent.

Step 1: Get the octal number.

            314

Step 2: Write the corresponding weights.

            3 = 8²  ;  1 = 8¹  ;  4 = 8⁰

Step 3: Multiply each digit in the given number with the corresponding weight to obtain product numbers.

             3*8² = 192  ;  1*8¹ = 8  ;  4*8⁰ = 4

Step 4: Add all the product numbers to get the decimal equivalent.

            192 + 8 + 4 = 204

Therefore; (314)₈ = (204)₁₀

Convert the hex number (4C8.2)₈ into its equivalent decimal number.

Step 1: Note down the given number.

            4C8.2

Step 2: Write down the corresponding weights.

            4 = 16²  ;  C = 16¹  ;  8 = 16⁰  ;  2 = 16^-1

Step 3: Multiply each digit in the given number with the corresponding weight to product numbers.

            4*16² = 1024  ;  C*16¹ = 192  ;  8*16⁰ = 8  ;  2*16^-1 = 0.125

Step 4: Add all the product numbers to get the decimal number.
            1024 + 192 + 8 + 0.125 = 1224.125

Therefore; (4C8.2)₈ = (1224.125)₁₀

Conversion from Decimal to Other Systems:

1. If the given decimal number consists of a decimal, then we have to first separate out the integer and fractional parts.
2. Then convert them differently to the desired base, and combine the converted parts to obtain the complete converted number.
3. The procedures for converting the integer part and the fractional part are completely different each other.

Steps to be followed:

• Step 1: Divide the integer  part of given decimal number by the base, noting the remainder.
• Step 2: Continue to divide  the quotient by the base until there is nothing left, keeping track of the remainders from each step.
• Step 3: List the remainder values in reverse order to find its equivalent.

Convert the decimal number (105)₁₀ to the equivalent binary number.

Figure below shows a simpler method. We divide the given number by the base of binary system.

Therefore; Binary number = 1101001
Thus; (105)₁₀ = (1101001)₂

Convert the decimal number (259)₁₀ into its hex equivalent.

The conversion takes place as follows:
The base is 16 so we divide the given number by 16.

Therefore; Hexadecimal number = 103
Thus; (259)₁₀ = (103)₁₆

Convert the decimal number (204)₁₀ into its equivalent octal number.

We divide the decimal number by the base of octal system which is 8.

Therefore; Octal number = (314)₈
Thus; (204)₁₀ = (314)₈